A Luttinger Hamiltonian is not enough
Download
Report
Transcript A Luttinger Hamiltonian is not enough
Wavepacket dynamics for Massive
Dirac electron
C.P. Chuu
Q. Niu
Dept. of Physics
Ming-Che Chang
Semiclassical electron dynamics in solid
(Ashcroft and Mermin, Chap 12)
dk
e
eE r B
dt
c
dr 1 E
dt
k
• Lattice effect hidden in E(k)
• Derivation is non-trivial
Explains • oscillatory motion of an electron in a DC field
(Bloch oscillation, quantized energy levels are known as
Wannier-Stark ladders)
• cyclotron motion in magnetic field
(quantized orbits relate to de Haas - van Alphen
effect)
•…
Limits of validity
eEa
Eg2 / EF
c
Eg2 / EF
Negligible inter-band transition (one-band approximation)
“never close to being violated in a metal”
Semiclassical dynamics - wavepacket approach
1. Construct a wavepacket that is
localized in both the r and the k spaces.
rW
k W
2. Using the time-dependent variational
principle to get the effective Lagrangian
Leff (rc , kc ; rc , kc ) W i
H W
t
e
= kc R kc rc A rc E (rc , kc )
c
Berry connection
R(k ) i un
un
k
Magnetization energy
of the wavepacket
Wavepacket energy
E (r , k ) E0 (k ) e (r )
e
L (k ) B
2mc
Self-rotating angular momentum
L(k ) m W r rc v W
3. Using the Leff to get the equations
of motion
• Bloch energy E0 (k )
dk
e
eE r B
dt
c
dr 1 E
k ( k )
dt
k
• Berry curvature (1983), as an
effective B field in k-space
( k ) i
Anomalous velocity due
to the Berry curvature
( k ) R ( k )
E (r , k ) E0 (k ) e (r )
Three quantities required to
know your Bloch electron:
u
u
k
k
• Angular momentum
(in the Rammal-Wilkinson form)
e
L (k ) B
2mc
L (k )
m u
u
E0 H
i k
k
Ω(k) and L(k) are zero when there are both
• time-reversal symmetry
• lattice inversion symmetry
(assuming there is no SO coupling)
1
N
Single band
Multiple bands
Basic quantities
Basics quantities
E (k ) E0 (k ) e (r )
R( k ) u i
u
k
e
L (k ) B
2mc
1
(k ) R R
2
Dynamics
dk
e
eE r B
dt
c
dr E
k ( k )
dt k
H (r , k ) E0 (k ) e (r )
Rij (k ) ui i
uj
k
e
L(k ) B
2mc
Magnetization
1
F (k ) R R i R , R
2
Dynamics
Covariant
dk
e
eE r B derivative
dt
c
dr
iR , H k F
dt
k
d
i
H (r , k ) k R
dt
SO interaction
Chang and Niu, PRL 1995, PRB 1996
Sundaram and Niu, PRB 1999
Culcer, Yao, and Niu PRB 2005
Shindou and Imura, Nucl. Phys. B 2005
• Relativistic electron (as a trial case)
• Semiconductor carrier
Construction of a Dirac wave packet
E0 ( q) m c c
2 4
( q)mc 2
Plane-wave solution
i eik r ui ,
2 2 2
q
ui u j ij
w d 3 qa (q , t ) 1 (q , t ) 1 2 ( q , t ) 2 ,
3
2
2
2
d
q
|
a
(
q
,
t
)
|
1;
|
|
|
|
1
1
2
Center of mass
w r w rc and
2mC2
2
3
d
qq | a(q, t ) | qc
If p mc, then the negative-energy components
are not negligible.
x / mc (Compton wave length c )
This wave packet has a minimal size
a0 : c : ae 1010 :1012 :1015
Classical
electron radius
• Angular momentum of the wave packet
c2
L ( kc ) 2
kc kc ;
1
or
Lij
0
ui u j ,
0
= 1+( k/mc) 2
1
1 (v / c ) 2
Ref: K. Huang, Am. J. Phys. 479 (1952).
• Energy of the wave packet
r
r
e
ge (kc )
L( kc )
2mc
2 mc 2
H(rc , kc ) E0 (kc ) e (rc ) M(kc ) B
M ( kc )
The self-rotation gives the correct
magnetic energy with g=2 !
• Gauge structure (gauge potential and gauge field,
or Berry connection and Berry curvature)
SU(2) gauge potential
c2
R
k
2 ( 1)
SU(2) gauge field
c2
c2
F 3
k k
2
1
Ref: Bliokh, Europhys. Lett. 72, 7 (2005)
Semiclassical dynamics of Dirac electron
• Precession of spin (Bargmann, Michel, and Telegdi, PRL 1959)
dS
e
1
k
B
E
S
dt mc
1
mc
S
2
L
• Center-of-mass motion
To liner fields >
For v<<c
dk
e
eE v B
dt
c
dr
k e
e
E F c k BF
dt m
2
k
B
ec
B
E
1
m mc 2
2
(B e / 2mc)
Or,
++++++++++
Spin-dependent
transverse velocity
L
---------for 1 GeV in 1 cm
L E (c )
106 !
2
L
mc
m*/ m
2
2
k m*r
m E , where m * c mc +B B
c
g ( e)
“hidden momentum”
m
S
2mc
Shockley-James paradox
(Shockley and James, PRLs 1967)
A simpler version (Vaidman, Am. J. Phys. 1990)
A charge and a solenoid:
q
S
B
E
Resolution of the paradox
• Penfield and Haus, Electrodynamics of Moving Media, 1967
• S. Coleman and van Vleck, PR 1968
A stationary current loop in an E field
Smaller m
m
Gain
energy
Lose
energy
E
Power flow and
momentum flow
// m E
Larger m
Force on a magnetic dipole
(Jackson, Classical Electrodynamics,
the 3rd ed.)
• magnetic charge model
( m B )
• current loop model
( m B )
d m E
dt c
Energy of the wave packet
H(rc , kc ) E0 (kc ) e (rc ) M(kc ) B
Where is the spin-orbit coupling energy?
Re-quantizing the semiclassical theory:
Effective Lagrangian (general)
(Chuu, Chang, and Niu, to be
published. Also see Duvar,
Horvath, and Horvath, Int J Mod
Phys 2001)
e
(Non-canonical variables)
kc R kc rc A rc E (rc , kc )
t
c
df
Standard form (canonical var.) ri , p j ij
=i †
p r E (r , p )
t
dt
Leff i †
Conversely, one can write
(correct to linear field)
new “canonical” variables,
r rc R(kc ) G (kc );
rc r R ( ) G ( );
e
e
A( rc ) B R(kc ),
c
2c
where G 1/ 2(R / k ) ( R B)
e
e
A(r ) B R( ),
c
c
where p e / cA(r )
p kc
kc p
(generalized Peierls substitution)
For Dirac electron, to linear order in fields
R,
r
1
R
k S
2 2
2m c
This is the SO interaction with the correct
= ( r ) c E k S
Thomas factor!
2mc
(r R) (r )
(Ref: Shankar and Mathur, PRL 1994)
Relativistic Pauli equation
Pair production
Dirac Hamiltonian (4-component)
e
H D c p A( r ) mc 2 e ( r )
c
Foldy-Wouthuysen
transformation Silenko, J.
Semiclassical energy
H(rc , kc ) E0 (kc ) e (rc ) M(kc ) B
Math. Phys. 44, 2952 (2003)
generalized Peierls substitution
rc rˆ R (ˆ ) G (ˆ );
p e / cA(r )
e
kc ˆ B R (ˆ ).
c
H P U † H DU
Pauli Hamiltonian (2-component)
H P ( )mc 2
B
E B B e (r )
( )[ ( ) 1]mc
( )
correct to first order in fields,
exact to all orders of v/c!
Ref: Silenko, J. Math. Phys. 44, 1952 (2003)
Why heating a cold pizza?
advantages of the wave packet approach
A coherent framework for
A heuristic model of the electron spin
Dynamics of electron spin precession (BMT)
Trajectory of relativistic electron (Newton-Wigner, FW )
Gauge structure of the Dirac theory, SO coupling (Mathur + Shankar)
Canonical structure, requantization (Bliokh)
2-component representation of the Dirac equation (FW, Silenko)
Also possible: Dirac+gravity, K-G eq, Maxwell eq…
Pair production
Relevant fields
Relativistic beam dynamics
Relativistic plasma dynamics
Relativistic optics
…
• Relativistic electron (as a trial case)
• Semiconductor carrier
Hall effect (E.H. Hall, 1879)
(Extrinsic) Spin Hall effect
(J.E. Hirsch, PRL 1999,
Dyakonov and Perel, JETP 1971.)
• skew scattering by spinless impurities
• no magnetic field required
Intrinsic spin Hall effect in p-type semiconductor
(Murakami, Nagaosa and Zhang, Science 2003; PRB 2004)
Valence band of GaAs:
Luttinger Hamiltonian (1956)
(for j=3/2 valence bands)
2
1
5 2
H
1 2 k 2 2 k J
2m
2
kˆ J (helicity)
is a good quantum number
(Non-Abelian) gauge potential
R ' (k ) u i
u '
k
Berry curvature,
due to monopole field in kspace r r
7 k
( k ) 2 2
4 k2
dk
eE
dt
dx E ( k ) dk
(k )
dt
dt
k
F
IJ
G
H K
Emergence of curvature by projection
Non-Abelian
• Free Dirac electron
Curvature for the whole space
F dR iR R 0
Curvature for a subspace
F d ( PRP ) iPRP PRP 0
• 4-band Luttinger
z
model (j=3/2)
Analogy in geometry
u
Ref: J.E. Avron, Les Houches 1994
x
v
y
Berry curvature in conduction band?
8-band Kane model
Rashba system (in asymm QW)
p2 r r
H
p z
2m
Is there any curvature simply
by projection?
There is no curvature anywhere
except at the degenerate point
(k ) (k )
8-band Kane model
Efros and Rosen, Ann. Rev. Mater. Sci. 2000
Gauge structure in conduction band
• Gauge potential, correct to k1
Eg
V2 1
1
k , V
R
2
2
3 Eg E
g
• Angular momentum, correct to k0
S Px X / m0
2m0V 2 1
1
L
,
3
E
E
g
g
Gauge structures and angular momenta in other subspaces
Chang et al, to be published
Re-quantizing the semiclassical theory:
generalized Peierls substitution:
Effective Hamiltonian
H (r , k ) E0 (k ) e (r ) eE R (k )
rc r R ( ) G ( );
e
e
kc p A(r ) B R( ),
c
c
where p e / cA(r )
ri , p j ij
Ref: Roth, J. Phys. Chem. Solids
1962; Blount, PR 1962
(rc ) (r ) E R
E0 (kc ) E0 ( p)
E
e
B L( k ) 2R m 0
2mc
p
• vanishes near band edge
e E0
B R
c p
• higher order in k
Spin-orbit coupling for conduction electron
eE R E k ,
• Same form as Rashba
eV 2 1
1
where
2
2
3 Eg E
g
( = 0 if 0 )
• In the absence of BIA/SIA
Ref: R. Winkler, SO coupling
effect in 2D electron and hole
systems, Sec. 5.2
Effective Hamiltonian for semiconductor carrier
q
B L( k )
2mc
g
H c (r , k ) E0 (k ) E k B B
2
H H (r , k ) E0 (k , J ) H E J k 2 H B B J
H (r , k ) E0 (k ) qE R (k )
H SO (r , k ) E0 (k ) SO E k 2 SO B B
Spin part
orbital part
2
eV 2 1
1
4
mV
1
1
, g 2
3 Eg2 E 2
3 2 Eg Eg
g
eV 2 1
1 4 mV 2 1
H
, H
3 Eg2
2 3 2 Eg
SO
Yu and Cardona,
Fundamentals of
semiconductors,
Prob. 9.16
eV 2
1
1 4 mV 2 1
, SO
3 E 2
2 3 2 Eg
g
Effective H’s agree with Winkler’s obtained using LÖwdin partition
Covered in this talk:
• Wave packet dynamics in multiple bands
• Relativistic electron
• Spin Hall effect
Not covered
• Wave packet dynamics in single band
• Anomalous Hall effect
• Quantum Hall effect
• (Anomalous) Nernst effect
• optical Hall effect
Forward jump and “side jump”
Berger and Bergmann, in The Hall effect and
its applications, by Chien and Westgate (1980)
(Picht 1929+Goos and Hanchen1947, Fedorov 1955+Imbert 1968,
Onoda, Murakami, and Nagaosa, PRL 2004; Bliokh PRL 2006)
• wave packet in BEC
(Niu’s group: Demircan, Diener, Dudarev, Zhang… etc )
Not related:
• thermal Hall effect
(Leduc-Righi effect, 1887)
• phonon Hall effect
(Strohm, Rikken, and Wyder, PRL 2005,
L. Sheng, D.N. Sheng, and Ting, PRL 2006)
Thank you !